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A characterization of weakly compact sets by convex functions
CHENG Qingjin *
Department of Mathematics，Xiamen University，XiaMen 361005
*Correspondence author
#Submitted by
Subject:
Funding: Specialized Research Fund for the Doctoral Program of Higher Education（No.200803841018）
Opened online:14 February 2012
Accepted by: none
Citation: CHENG Qingjin.A characterization of weakly compact sets by convex functions[OL]. [14 February 2012] http://en.paper.edu.cn/en_releasepaper/content/4462252
 
 
This note introduces a convexity property of convexfunctions defined on a nonempty convex set in Banach spaces and establishes several equivalent characterizations of weakcompactly subsets of Banach spaces by the property. More precisely, let X be a Banach space and let K∈X be a convex bounded subset. Then (i) If X is separable, then K is weakly compact iff thereexists a continuous 2R convex function on K.(ii) If X is nonseparable then K is weakly compact iffthere exists a continuous w2R convex function on K.
Keywords:Banach space; weakly compact set; convex function
 
 
 

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